Abstract
Full Text
MATHEMATICS
S. M. CHASHCHNIKOV
FIELD THEORY OF LOCAL HYPERCONES IN \(X_n\)
(Presented by Academician I. G. Petrovskii, 11 VI 1957)
- The field theory of local conical surfaces in \(X_n\) is of interest in connection with applications of differential geometry to the calculus of variations and to the theory of first-order partial differential equations with one unknown function. The field theory of local conical surfaces was first constructed in three-dimensional metric Euclidean space by C. Lee and G. Scheffers \((^3)\). Later this theory was considered by V. V. Wagner \((^4)\). From the point of view of applications, the assumption that the space is metric Euclidean is not expedient, since the study of a variational problem and of a differential equation specified up to arbitrary transformations of variables by means of a group of motions is of no special interest.
In 1948 V. V. Wagner \((^1)\) constructed the field theory of local conical surfaces in \(X_3\) and considered its applications to the theory of differential equations and to the calculus of variations. In the present work the field theory of local hypercones in \(X_n\) for \(n>4\) is considered.
- A central hypercone in a central \(E_n\) is uniquely determined by specifying an \((n-2)\)-dimensional surface that is its director. It is obvious that the director is not determined uniquely. A hypercone is called pseudoregular if it admits, as its director, a regular \((n-2)\)-dimensional surface. We shall consider only pseudoregular hypercones.
It is expedient to specify the director by parametric equations
\[ x^\alpha = l^\alpha(\eta^a) \quad (a, b, \ldots, e = 1, \ldots, n-2). \tag{1} \]
Assuming that the parameters \(\eta^a\) may be subjected to transformations
\[ \eta'^a = \varphi^a(\eta^e), \tag{2} \]
where \(\varphi^a\) are regular functions of class \(v\), we may regard the director as an \((n-2)\)-dimensional Weyl–Whitehead space \(X_{n-2}\). In this \(X_{n-2}\) the \(W\)-tensor densities \(\mathfrak{G}_{ba}, \mathfrak{A}_{cba}\) of weight \(-\dfrac{2}{n-2}\), the tensor \(h_{ba}\), and the object of affine connection \(G^c_{ba}\) are defined invariantly.
As V. V. Wagner \((^5)\) showed, an \((n-2)\)-dimensional regular surface in a central \(E_n\) is determined, up to an arbitrary centro-affine transformation, by specifying in \(X_{n-2}\), for \(n>4\), the \(W\)-tensor densities \(\mathfrak{G}_{ba}, \mathfrak{A}_{cba}\) and the contracted object of affine connection \(G_a\), while for \(n=4\) the tensor \(h_{ba}\) is also adjoined to them.
A director of a hypercone for which the conditions
\[ \nabla_a \mathfrak{M} = 0, \tag{3} \]
are satisfied,
where \(\mathfrak M\) is a scalar \(W\)-density of weight \(\dfrac{2}{n-2}\), which is a function of the \(W\)-tensor densities \(\mathfrak G_{ba}\) and \(\mathfrak A_{cba}\), we shall call a normalized with the aid of the \(W\)-density \(\mathfrak M\). Conditions (3) define a one-parameter family of normalized directrices obtained from one another by means of a similarity transformation with center of similitude at the center \(E_n\). It is not hard to see that, for a normalized directrix, the relations
\[ G_a=\partial_a \ln \mathfrak M, \tag{4} \]
hold, whence follows the theorem:
Theorem 1. A central pseudoregular hypercone in a central \(E_n\) is determined, up to an arbitrary central-affine transformation, by prescribing in \(X_{n-2}\), for \(n>4\), the \(W\)-tensor densities \(\mathfrak G_{ba}\) and \(\mathfrak A_{cba}\), and for \(n=4\) the tensor \(h_{ba}\) is added to them.
- Defining local hypercones of a field by prescribing their normalized directrices, we reduce the field theory of local hypercones in \(X_n\) to the field theory of local \((n-2)\)-dimensional normalized surfaces
\[ x^\alpha=l^\alpha(\xi^\beta,\eta^e), \tag{5} \]
given up to arbitrary local similarity transformations
\[ ' x^\alpha=\frac{x^\alpha}{\sigma(\xi^\beta)} . \tag{6} \]
Assuming that the admissible transformations of the parameters \(\eta^a\) are determined by the equations
\[ ' \eta^\alpha=\varphi^a(\xi^\alpha,\eta^e) \left(\det\left\|\frac{\partial \varphi^a}{\partial \eta^b}\right\|\ne 0\right), \tag{7} \]
where \(\varphi^a\) are arbitrary functions, continuously differentiable a sufficient number of times with respect to the variables \(\xi^\alpha\) and \(\eta^a\), we may regard this field as a composite manifold \(X_{n-2}(X_n)\) with the linear affine connection (2). In this \(X_{n-2}(X_n)\), \(n\) fields of \(W\)-densities \(\mathfrak n_a, m, \mathfrak n\) of weights respectively \(-\dfrac{2}{n-2},\,0,\,-\dfrac{2}{n-2}\) are invariantly defined:
\[ \mathfrak n_a=\mathfrak n_{a\alpha}(\xi^\beta,\eta^e)d\xi^\alpha;\qquad m=m_\alpha(\xi^\beta,\eta^e)d\xi^\alpha;\qquad \mathfrak n=\mathfrak n_\alpha(\xi^\beta,\eta^e)d\xi^\alpha . \tag{8} \]
These \(W\)-densities satisfy the following system of differential equations in total differentials:
\[ \overset{2}{\nabla}_b \mathfrak n_a=-\mathfrak G_{ba}m+v_{ba}\mathfrak n;\qquad \overset{2}{\nabla}_a m=\mathfrak H^{\,b}_{\ .a}\mathfrak n_b+\mathfrak W_a\mathfrak n;\qquad \overset{2}{\nabla}_a \mathfrak n=\mathfrak n_a, \tag{9} \]
where \(\mathfrak H^{\,b}_{\ .a}=\mathfrak G^{bc}h_{ca}\); \(\overset{2}{\nabla}_c\) is the symbol of covariant differentiation with respect to the connection with coefficients
\[ \overset{2}{G}{}^{c}_{ba}=G^c_{ba}+\mathfrak G^{cd}\mathfrak A_{bad}, \tag{10} \]
and \(v_{ba}\) and \(\mathfrak W_a\) are known functions of the \(W\)-tensor densities \(\mathfrak G_{ba}\) and \(\mathfrak A_{cba}\). Under local similarity transformations (6), the \(W\)-densities \(\mathfrak n_a\), \(m\), and \(\mathfrak n\) are multiplied by \(\sigma(\xi^\alpha)\).
Suppose that in \(X_{n-2}(X_n)\), by means of the Pfaff equations
\[ \delta\eta^a=d\eta^a+\Gamma^a=0 \qquad (\Gamma^a=\Gamma^a_\alpha(\xi^\beta,\eta^e)d\xi^\alpha) \tag{11} \]
a linear connection, invariant with respect to the local similarity transformations (6), is defined, and let
\[ \Gamma^\alpha=\nu^{ba}\eta_b+\nu_{n-1}^{a}m+\nu_n^a\eta; \tag{12} \]
\[ [\partial\eta_e]=\mathfrak{R}_e^{\cdot ba}[\eta_b\eta_a]+2\mathfrak{R}_{e\,(n-1)}^{\cdot a}[\eta_a m]+2\mathfrak{R}_{e\,(n)}^{\cdot a}[\eta_a\eta]+2\mathfrak{R}_{e\,(n-1,n)}[m\eta]; \tag{13} \]
\[ [\partial\eta]=\Lambda^{ba}[\eta_b\eta_a]+2\Lambda_{(n-1)}^a[\eta_a m]+2\Lambda_{(n)}^a[\eta_a\eta]+2\Lambda_{(n-1,n)}[m\eta], \tag{14} \]
where the square brackets denote Cartan exterior multiplication. Introducing the notation
\[ [D\eta_e]=\mathfrak{B}_e^{\cdot ba}[\eta_b\eta_a]+2W_{e\,(n-1)}^{\cdot a}[\eta_a m]+2\mathfrak{B}_{e\,(n)}^{\cdot a}[\eta_a\eta]+2W_{e\,(n-1,n)}[m\eta]; \tag{15} \]
\[ [Dm]=J^{ba}[\eta_b\eta_a]+2J_{n-1}^{a}[\eta_a m]+2J_n^a[\eta_a\eta]+2J_{(n-1,n)}[m\eta]; \tag{16} \]
\[ [D\eta]=\mathfrak{B}^{ba}[\eta_b\eta_a]+2V_{n-1}^{a}[\eta_a m]+2\mathfrak{B}_n^a[\eta_a\eta]+2V_{(n-1,n)}[m\eta], \tag{17} \]
where \(D\) is the symbol of the absolute basis differential, we define the invariant connection by the conditions:
\[ \mathfrak{B}^{ba}=0;\qquad V_{n-1}^{a}=0;\qquad W_c=0;\qquad D_{n-1}\mathfrak{G}^{ba}=0;\qquad D_{n-1}\mathfrak{M}=0, \tag{18} \]
where the differential operator \(D_{n-1}\) is defined by the symbolic equality
\[ D=\eta_a\vartheta^a+mD_{n-1}+\eta\vartheta_n. \tag{19} \]
After some computations we obtain
\[ \nu^{ba}=\Lambda^{ba}+\frac12\,\iota^\omega(\partial_\omega\mathfrak{G}^{ba}-\mathfrak{G}^{ba}\partial_\omega\ln\mathfrak{M})+\Lambda_{(n-1)}^e\partial_c\mathfrak{G}^{ba}-\mathfrak{G}^{ba}\partial_c\ln\mathfrak{M}; \tag{20} \]
\[ \nu_n^a=-2\Lambda_{(n-1)}^a; \tag{21} \]
\[ \nu_n^a=2\mathfrak{G}^{aa}_{(n-1,n)}\left(\mathfrak{R}_c+\Lambda_{(n-1)}^e\vartheta_{ec}\right). \tag{22} \]
The invariant connection can be defined not in a unique way, at least because the \(W\)-density \(\mathfrak{M}\) is chosen with great arbitrariness.
- If the field of local hypercones in \(X_n\) is constant in some domain \(T\) of the basis \(X_n\), then there exists such a system of coordinates in \(X_n\), a geometric domain containing \(T\), and such a field of local coordinate systems in this domain that the equations of the normalized directions of the local hypercones of the field in the domain \(T\) are written in the form
\[ x^\alpha=\delta(\xi^\beta)l^\alpha(\eta^\varepsilon). \tag{23} \]
Theorem 2. The field of local \((n-2)\)-dimensional normalized surfaces in \(X_n\), for \(n\geqslant 4\), can be transformed in the domain \(T\) by means of local similarity transformations into a constant field if and only if in the domain \(T\) the conditions
\[ [\partial\Gamma^a]-[\Gamma^e\partial_e\Gamma^a]=0;\qquad D\mathfrak{G}_{ba}=0;\qquad D\mathfrak{R}_{cba}=0; \tag{24} \]
are satisfied, and for \(n=4\) also the condition
\[ Dh_{ba}=0. \tag{25} \]
The necessity of these conditions may be regarded as obvious. To prove sufficiency, choose local coordinate systems in the domain \(T\) in such a way that all the Pfaffians of the connection \(\Gamma^a\) vanish. It is not difficult to see that from conditions (24) and (25) it follows that the \(W\)-tensor densities \(\mathfrak{S}_{ba}\), \(\mathfrak{A}_{cba}\), and the tensor \(h_{ba}\) in the domain \(T\) do not depend on the coordinates \(\xi^\alpha\) of the point of the base \(X_n\). Therefore the solution of system (9) can be written in the form
\[ \underset{\alpha}{\mathfrak{n}}_a = \underset{\alpha}{*\mathfrak{n}}_a(\eta^e)\, \overset{\alpha}{\omega}(\xi^\beta); \qquad \underset{\alpha}{m} = \underset{\alpha}{*m}(\eta^e)\, \overset{\alpha}{\omega}(\xi^\beta); \qquad \underset{\alpha}{\mathfrak{n}} = \underset{\alpha}{*\mathfrak{n}}(\eta^e)\, \overset{\alpha}{\omega}(\xi^\beta), \tag{26} \]
where \(\bigl(\underset{\alpha}{*\mathfrak{n}}_a,\underset{\alpha}{*m},\underset{\alpha}{*\mathfrak{n}}\bigr)\) are \(n\) linearly independent solutions of system (9), and
\[ \overset{\alpha}{\omega}=e^\alpha_\beta(\xi^\gamma)\,d\xi^\beta . \]
It may be shown that
\[ \overset{\alpha}{\omega}=\theta_{(\alpha)}\,d\varphi^{(\alpha)}; \qquad \theta_\alpha=\sigma(\xi^\beta)\psi_\alpha(\varphi^\alpha), \tag{27} \]
where \(\varphi^\alpha\) is a function of the variables \(\xi^\beta\), and summation over the index \(\alpha\) is not implied. Introducing in \(X_n\) the new variables
\[ \widetilde{\xi}^{\,\alpha} = \int \psi_{(\alpha)}(\varphi^\alpha)\,d\varphi^{(\alpha)}, \tag{28} \]
we obtain
\[ \mathfrak{n}_a = \sigma\,\underset{\alpha}{*\mathfrak{n}}_a\,d\widetilde{\xi}^{\,\alpha}; \qquad m = \sigma\,\underset{\alpha}{*m}\,d\widetilde{\xi}^{\,\alpha}; \qquad \mathfrak{n} = \sigma\,\underset{\alpha}{*\mathfrak{n}}\,d\widetilde{\xi}^{\,\alpha}, \tag{29} \]
whence it follows that, in the new coordinate system, the equations of the \((n-2)\)-dimensional normalized surfaces of the field will have the form
\[ l^\alpha(\widetilde{\xi}^{\,\beta},\eta^e) = \frac{1}{\sigma(\xi^\beta)}\,l^\alpha(\eta^e). \tag{30} \]
Applying the results obtained to the field of local hypercones in \(X_n\), we shall have:
Theorem 3. The field of local hypercones in \(X_n\) for \(n\geqslant 4\) will be constant in the domain \(T\) of the base \(X_n\) if and only if in the domain \(T\) conditions (24) are satisfied, and for \(n=4\) also condition (25).
The author expresses his gratitude to V. V. Wagner, under whose supervision the present work was carried out.
Saratov State University
named after N. G. Chernyshevsky
Received
8 X 1956
References
- V. V. Wagner, Transactions of the Seminar on Vector and Tensor Analysis, vol. 6, 257 (1948).
- V. V. Wagner, Transactions of the Seminar on Vector and Tensor Analysis, vol. 8, 11 (1950).
- S. Lie, G. Scheffers, Geometrie der Berührungstransformationen, 1, 1896.
- V. Wagner, Mathematical Collection, 8, 3 (1940).
- V. Wagner, Ann. of Math., 49 (1), 141 (1948).